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Article

# Important Criteria for Asymptotic Properties of Nonlinear Differential Equations

by 1,†, 2,3,† and 4,5,6,*,†
1
Department of Computer Engineering, College of Computers and Information Technology, Taif University, Taif 21944, Saudi Arabia
2
Section of Mathematics, International Telematic University Uninettuno, CorsoVittorio Emanuele II, 39, 00186 Roma, Italy
3
4
Research Center for Quantum Technology, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
5
Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
6
Athens Institute for Education and Research, Mathematics and Physics Divisions, 8 Valaoritou Street, Kolonaki, 10671 Athens, Greece
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2021, 9(14), 1659; https://doi.org/10.3390/math9141659
Received: 7 June 2021 / Revised: 1 July 2021 / Accepted: 10 July 2021 / Published: 14 July 2021

## Abstract

:
In this article, we prove some new oscillation theorems for fourth-order differential equations. New oscillation results are established that complement related contributions to the subject. We use the Riccati technique and the integral averaging technique to prove our results. As proof of the effectiveness of the new criteria, we offer more than one practical example.

## 1. Introduction

In this manuscript, we are concerned with the asymptotic behavior of solutions to fourth-order differential equations:
$m z Ψ r 1 ς ″ ′ z ′ + ω ˜ z Ψ r 2 δ α z = 0 ,$
where $Ψ r i [ s ] = | s | r i − 1 s , i = 1 , 2 , ς z = δ z + y ˜ z δ α ˜ z , m , y ˜ , ω ˜ ∈ C [ z 0 , ∞ ) , m z > 0 , m ′ z ≥ 0 , ω ˜ z > 0 , 0 ≤ y ˜ z < y ˜ 0 < ∞ , α ˜ , α ∈ C [ z 0 , ∞ ) , α ˜ z ≤ z , lim z → ∞ α ˜ z = lim z → ∞ α z = ∞$; and $r 1$ and $r 2$ are quotients of odd positive integers, under the assumption of the following:
$∫ z 0 ∞ 1 m 1 / r 1 s d s = ∞ .$
The theory of the oscillation of delay of differential equations is a fertile study area and has attracted the attention of many authors recently. This is due to the existence of many important applications of this theory in neural networks, biology, social sciences, engineering, etc.; see [1,2].
A study of the behavior of solutions to higher order differential equations yields much fewer results than for the least order equations although they are of the utmost importance in a lot of applications, especially neutral delay differential equations.
Currently, there are studies on the oscillation results of differential equations, so many of these studies have been devoted to study the oscillation of different classes of differential equations by using different techniques in order to establish sufficient conditions to ensure the oscillatory behavior of the solutions of (1), see [3,4,5].
Xing et al. [6] presented criteria for oscillation of the equation as follows:
$m z ς n − 1 z r 1 ′ + ω ˜ z δ r 1 α z = 0 ,$
under the conditions
$α − 1 z ′ ≥ α 0 > 0 , α ˜ ′ z ≥ α ˜ 0 > 0 , α ˜ − 1 α z < z$
and
$lim inf z → ∞ ∫ α ˜ − 1 α z z ω ˜ ^ s m s s n − 1 r 1 d s > 1 α 0 + y ˜ 0 r 1 α 0 α ˜ 0 > n − 1 ! r 1 e ,$
where $0 ≤ y ˜ z < y ˜ 0 < ∞$ and $ω ˜ ^ z : = min ω ˜ α − 1 z , ω ˜ α − 1 α ˜ z$. Moreover, the authors used the comparison method to obtain oscillation conditions for this equation.
Bazighifan et al. [7] presented oscillation results for the following fourth-order equation:
$m z ς ′ ′ ′ z r 1 ′ + ω ˜ z δ r 1 α z = 0 ,$
under the conditions
$∫ z 0 ∞ 1 m 1 / r 1 s d s < ∞$
using the Riccati technique.
Zhang et al. [8] established oscillation criteria for the following equation:
$m z ς n − 1 z r 1 ′ + ω ˜ z f δ α z d s = 0$
and under the condition
$∫ z 0 ∞ k ρ z E z − 1 4 λ ρ ′ z ρ z 2 η z d s = ∞ .$
Chatzarakis et al. [9], by using the Riccati technique, established asymptotic behavior for the following neutral equation:
$m z ς ″ ′ z r 1 ′ + ∫ a b ω ˜ z , s f δ α z , s d s = 0 .$
The authors in [6,7] used the comparison technique that differs from the one we used in this article. Their approach is based on using these mentioned methods to reduce Equation (1) into a first-order equation, while in our article, we discuss the oscillatory properties of differential equations with a middle term and with a canonical operator of the neutral-type, and we employ a different approach based on using the integral averaging technique and the Riccati technique to reduce the main equation into a first-order inequality to obtain more effective oscillatory properties.
The purpose of this article is to establish new oscillation criteria for (1). The methods used in this paper simplify and extend some of the known results that are reported in the literature [6,7]. The authors in [6,7] used a comparison technique that differs from the one we used in this article.

## 2. Oscillation Criteria

We next present the lemmas needed for the proof of the original results:
Lemma 1
([10]). If $δ ( i ) z > 0 ,$$i = 0 , 1 , … , n ,$ and $δ n + 1 z < 0 ,$ then the following holds:
$n ! δ z z n ≥ n − 1 ! δ ′ z z n − 1 .$
Lemma 2
([11]). Let $δ ∈ C n z 0 , ∞ , 0 , ∞ .$ Assume that $δ n z$ is of fixed sign and not identically zero on $z 0 , ∞$ and that there exists a $z 1 ≥ z 0$ such that $δ n − 1 z δ n z ≤ 0$ for all $z ≥ z 1$. If $lim z → ∞ δ z ≠ 0 ,$ then for every $μ ∈ 0 , 1$ there exists $z μ ≥ z 1$ such that the following holds:
$δ z ≥ μ n − 1 ! z n − 1 δ n − 1 z f o r z ≥ z μ .$
Lemma 3
([12]). Let $a ≥ 0$; then, the following holds:
$X δ − Y δ a + 1 / a ≤ a a ( a + 1 ) − a + 1 Y − a X a + 1 ,$
where $Y > 0$ and X are constants.
Lemma 4
([13]). Assume that $δ z$ is an eventually positive solution of Equation (1). Then,
$C a s e N 1 : ς z > 0 , ς ′ z > 0 , ς ″ z > 0 , ς ″ ′ z > 0 , C a s e N 2 : ς z > 0 , ς ′ z > 0 , ς ″ z < 0 , ς ″ ′ z > 0 .$
Here are the notations used for our study:
$E 1 z = β z ω ˜ z 1 − y ˜ 0 r 2 A 1 r 2 − r 1 α z z 3 r 2 , Φ z = 1 − y ˜ 0 r 2 / r 1 h z A 2 r 2 / r 1 − 1 z ∫ z ∞ 1 m u ∫ u ∞ ω ˜ s α r 2 s s r 2 d s 1 / r 1 d u$
and
$Θ z = r 1 μ 1 z 2 2 m 1 / r 1 z β 1 / r 1 z .$
Lemma 5.
Let $δ z$ is an eventually positive solution of Equation (1), then
$m z ς ″ ′ z r 1 ′ ≤ − G z ς ″ ′ α z r 2 ,$
where
$G z = ω ˜ z 1 − y ˜ 0 r 2 μ 6 α z 3 r 2 .$
Proof.
Let $δ z$ is an eventually positive solution of Equation (1). From definition of $ς z = δ z + y ˜ z δ α ˜ z$, we obtain the following:
$δ z ≥ ς z − y ˜ 0 δ α ˜ z ≥ ς z − y ˜ 0 ς α ˜ z ≥ 1 − y ˜ 0 ς z ,$
which with (1), results in the following:
$m z ς ″ ′ z r 1 ′ + ω ˜ z 1 − y ˜ 0 r 2 ς r 2 α z ≤ 0 .$
Using Lemma 2, we see the following:
$ς z ≥ μ 6 z 3 ς ″ ′ z .$
Combining (4) and (5), we find the following:
$m z ς ″ ′ z r 1 ′ + ω ˜ z 1 − y ˜ 0 r 2 μ 6 α z 3 r 2 ς ″ ′ α z r 2 ≤ 0 .$
Thus, (3) holds. This completes the proof. □
Lemma 6.
Let $δ z$ is an eventually positive solution of Equation (1) and
$B ′ z ≤ β ′ z β z B z − E 1 z − r 1 μ 1 z 2 2 m 1 / r 1 z β 1 / r 1 z B r 1 + 1 r 1 z , i f ς s a t i s f i e s N 1$
and
$A ′ z ≤ − Φ z + h ′ z h z A z − 1 h z A 2 z , i f ς s a t i s f i e s N 2 ,$
where
$B z : = β z m z ς ″ ′ z r 1 ς r 1 z > 0$
and
$A z : = h z ς ′ z ς z , z ≥ z 1 .$
Proof.
Let $δ z$ is an eventually positive solution of Equation (1). Let $N 1$ holds. From (8) and (4), we find the following:
$B ′ z ≤ β ′ z β z B z − β z ω ˜ z 1 − y ˜ 0 r 2 ς r 2 α z ς r 1 z − r 1 β z m z ς ″ ′ z r 1 ς r 1 + 1 z ς ′ z .$
Using Lemma 1, we find
$ς z ≥ z 3 ς ′ z$
and hence,
$ς α z ς z ≥ α 3 z z 3 .$
It follows from Lemma 2 that
$ς ′ z ≥ μ 1 2 z 2 ς ″ ′ z ,$
for all $μ 1 ∈ 0 , 1$ and every sufficiently large z. Thus, by (10)–(12), we obtain the following:
$B ′ z ≤ β ′ z β z B z − β z ω ˜ z 1 − y ˜ 0 r 2 ς r 2 − r 1 z α z z 3 r 2 − r 1 μ 1 z 2 2 m 1 / r 1 z β 1 / r 1 z B r 1 + 1 r 1 z .$
Since $ς ′ z > 0$, there exist $z 2 ≥ z 1$ and $A 1 > 0$ such that the following holds:
$ς z > A 1 .$
Thus, we obtain the following:
$B ′ z ≤ β ′ z β z B z − β z ω ˜ z 1 − y ˜ 0 r 2 A r 2 − r 1 α z z 3 r 2 − r 1 μ 1 z 2 2 m 1 / r 1 z β 1 / r 1 z B r 1 + 1 r 1 z ,$
which yields the following:
$B ′ z ≤ β ′ z β z B z − E 1 z − r 1 μ 1 z 2 2 m 1 / r 1 z β 1 / r 1 z B r 1 + 1 r 1 z .$
Thus, (6) holds.
Let $N 2$ hold. Integrating (4) from z to u, we find the following:
$m u ς ″ ′ u r 1 − m z ς ″ ′ z r 1 ≤ − ∫ z u ω ˜ s 1 − y ˜ 0 r 2 ς r 2 α s d s .$
From Lemma 1, we obtain the following:
$ς z ≥ z ς ′ z$
and hence,
$ς α z ≥ α z z ς z .$
For (14), letting $u → ∞$ and using (15), we obtain the following:
$m z ς ″ ′ z r 1 ≥ 1 − y ˜ 0 r 2 ς r 2 z ∫ z ∞ ω ˜ s α r 2 s s r 2 d s .$
Integrating (16) from z to , we find the following:
$ς ′ ′ z ≤ − 1 − y ˜ 0 r 2 / r 1 ς r 2 / r 1 z ∫ z ∞ 1 m u ∫ u ∞ ω ˜ s α r 2 s s r 2 d s 1 / r 1 d u ,$
From the definition of $A z$, we see that $A z > 0$ for $z ≥ z 1 ,$ and using (13) and (17), we find the following:
$A ′ z = h ′ z h z A z + h z ς ″ z ς z − h z ς ′ z ς z 2 ≤ h ′ z h z A z − 1 h z A 2 z − 1 − y ˜ 0 r 2 / r 1 h z ς r 2 / r 1 − 1 z ∫ z ∞ 1 m u ∫ u ∞ ω ˜ s α r 2 s s r 2 d s 1 / r 1 d u .$
Since $ς ′ z > 0$, there exist $z 2 ≥ z 1$ and $A 2 > 0$ such that the following holds:
$ς z > A 2 .$
Thus, we obtain the following:
$A ′ z ≤ − Φ z + h ′ z h z A z − 1 h z A 2 z ,$
Thus, (7) holds. Proof of the theorem is completed. □
Definition 1.
Let
$D = { z , s ∈ R 2 : z ≥ s ≥ z 0 } and D 0 = { z , s ∈ R 2 : z > s ≥ z 0 } .$
The function $G i ∈ C D , R$ fulfills the following conditions:
(i)
$G i z , s = 0$ for $z ≥ z 0 , G i z , s > 0 , z , s ∈ D 0 ;$
(ii)
The functions $h , υ ∈ C 1 z 0 , ∞ , 0 , ∞$ and $g i ∈ C D 0 , R$ such that
$∂ ∂ s G 1 z , s + β ′ s β s G z , s = g 1 z , s G 1 r 1 / r 1 + 1 z , s$
and
$∂ ∂ s G 2 z , s + h ′ s h s G 2 z , s = g 2 z , s G 2 z , s .$
Now, we present some Philos-type oscillation criteria for (1).
Theorem 1.
Let (24) hold. If $β , h ∈ C 1 z 0 , ∞ , R$ such that
$lim sup z → ∞ 1 G z , z 1 ∫ z 1 z G z , s E 1 s − g 1 r 1 + 1 z , s G 1 r 1 z , s r 1 + 1 r 1 + 1 2 r 1 m s β s μ 1 s 2 r 1 d s = ∞$
for all $μ 2 ∈ 0 , 1 ,$ and
$lim sup z → ∞ 1 G 2 z , z 1 ∫ z 1 z G 2 z , s Φ s − h s g 2 2 z , s 4 d s = ∞ ,$
then (1) is oscillatory.
Proof.
Let $δ$ be a non-oscillatory solution of (1), we see that $δ > 0$. Assume that $N 1$ holds. Multiplying (6) by $G z , s$ and integrating the resulting inequality from $z 1$ to z, we obtain the following:
$∫ z 1 z G z , s E 1 s d s ≤ B z 1 G z , z 1 + ∫ z 1 z ∂ ∂ s G z , s + β ′ s β s G z , s B s d s − ∫ z 1 z Θ s G z , s B r 1 + 1 r 1 s d s .$
From (18), we obtain the following:
$∫ z 1 z G z , s E 1 s d s ≤ B z 1 G z , z 1 + ∫ z 1 z g 1 z , s G 1 r 1 / r 1 + 1 z , s B s d s − ∫ z 1 z Θ s G z , s B r 1 + 1 r 1 s d s .$
Using Lemma 3 with $V = Θ s G z , s , U = g 1 z , s G 1 r 1 / r 1 + 1 z , s$ and $δ = B s$, we obtain the following:
$g 1 z , s G 1 r 1 / r 1 + 1 z , s B s − Θ s G z , s B r 1 + 1 r 1 s ≤ g 1 r 1 + 1 z , s G 1 r 1 z , s r 1 + 1 r 1 + 1 2 r 1 m z β z μ 1 z 2 r 1 ,$
which, with (22) gives the following:
$1 G z , z 1 ∫ z 1 z G z , s E 1 s − g 1 r 1 + 1 z , s G 1 r 1 z , s r 1 + 1 r 1 + 1 2 r 1 m s β s μ 1 s 2 r 1 d s ≤ B z 1 ,$
Assume that $N 2$ holds. Multiplying (7) by $G 2 z , s$ and integrating the resulting inequality from $z 1$ to z, we find the following:
$∫ z 1 z G 2 z , s Φ s d s ≤ A z 1 G 2 z , z 1 + ∫ z 1 z ∂ ∂ s G 2 z , s + h ′ s h s G 2 z , s A s d s − ∫ z 1 z 1 h s G 2 z , s A 2 s d s .$
Thus,
$∫ z 1 z G 2 z , s Φ s d s ≤ A z 1 G 2 z , z 1 + ∫ z 1 z g 2 z , s G 2 z , s A s d s − ∫ z 1 z 1 h s G 2 z , s A 2 s d s ≤ A z 1 G 2 z , z 1 + ∫ z 1 z h s g 2 2 z , s 4 d s$
and so
$1 G 2 z , z 1 ∫ z 1 z G 2 z , s Φ s − h s g 2 2 z , s 4 d s ≤ A z 1 ,$
which contradicts (21). Proof of the theorem is completed. □
Corollary 1.
Let (24) hold. If $β , h ∈ C 1 z 0 , ∞ , R$ such that
$∫ z 0 ∞ E 1 s − 2 r 1 r 1 + 1 r 1 + 1 m s β ′ s r 1 + 1 μ 1 r 1 s 2 r 1 β r 1 s d s = ∞$
and
$∫ z 0 ∞ Φ s − h ′ s 2 4 h s d s = ∞ ,$
for some $μ 1 ∈ 0 , 1$ and every $A 1 , A 2 > 0$, then (1) is oscillatory.

## 3. Example

This section presents some interesting examples to examine the applicability of theoretical outcomes.
Example 1.
Consider the following equation:
$δ + 1 2 δ 1 3 z 4 + ω ˜ 0 z 4 δ 1 2 z = 0 , z ≥ 1 , ω ˜ 0 > 0 .$
Let $r 1 = r 2 = 1 ,$$m z = 1 ,$$y ˜ z = 1 / 2 ,$$α ˜ z = z / 3 ,$$α z = z / 2$ and $ω ˜ z = ω ˜ 0 / z 4$. Hence, it is easy to see that
$∫ z 0 ∞ 1 m 1 / r 1 s d s = ∞ , E 1 z = ω ˜ 0 16 s$
and
$Φ z : = ω ˜ 0 24 .$
If we put $β s : = z 3$ and $h z : = z 2$, then we find the following:
$∫ z 0 ∞ E 1 s − 2 r 1 r 1 + 1 r 1 + 1 m s β ′ s r 1 + 1 μ 1 r 1 s 2 r 1 β r 1 s d s = ∫ z 0 ∞ ω ˜ 0 16 s − 9 2 μ 1 s d s$
and
$∫ z 0 ∞ Φ s − h ′ s 2 4 h s d s = ∫ z 0 ∞ ω ˜ 0 24 − 1 d s .$
Thus,
$ω ˜ 0 > 72$
and
$ω ˜ 0 > 24 .$
From Corollary 1, Equation (25) is oscillatory if $ω ˜ 0 > 72 .$
Example 2.
Consider the following equation:
$z δ + y ˜ 0 δ γ z ″ ′ ′ + ω ˜ 0 z 3 δ η z = 0 , z ≥ 1 ,$
where $y ˜ 0 ∈ 0 , 1 , γ , η ∈ 0 , 1$ and $ω ˜ 0 > 0$. Let $r 1 = r 2 = 1 ,$$m z = z ,$$y ˜ z = y ˜ 0 ,$$α ˜ z = γ z ,$$α z = η z$ and $ω ˜ z = ω ˜ 0 / z 3$. Hence, if we set $β s : = z 2$ and $h z : = z$, then we get
$E 1 z = ω ˜ 0 1 − y ˜ 0 η 3 z , Φ z = ω ˜ 0 1 − y ˜ 0 η 4 z .$
Thus, (23) and (24) become the following:
$∫ z 0 ∞ E 1 s − 2 r 1 r 1 + 1 r 1 + 1 m s β ′ s r 1 + 1 μ 1 r 1 s 2 r 1 β r 1 s d s = ∫ z 0 ∞ ω ˜ 0 1 − y ˜ 0 η 3 s − 2 μ 1 s d s$
and
$∫ z 0 ∞ Φ s − h ′ s 2 4 h s d s = ∫ z 0 ∞ ω ˜ 0 1 − y ˜ 0 η 4 s − 1 4 s d s .$
So,
$ω ˜ 0 > 2 1 − y ˜ 0 η 3$
and
$ω ˜ 0 > 1 1 − y ˜ 0 η .$
From Corollary 1, Equation (28) is oscillatory if (29) holds.

## 4. Conclusions

In this work, we prove some new oscillation theorems for (1). New oscillation results are established that complement related contributions to the subject. We used the Riccati technique and integral averages technique to obtain some new results to the oscillation of Equation (1) under the condition $∫ z 0 ∞ 1 m 1 / r 1 s d s = ∞ .$ In future work, we will study this type of equation under the following condition:
$∫ z 0 ∞ 1 m 1 / r 1 s d s < ∞ ,$
We also introduce some important oscillation criteria of differential equations of the fourth-order and under the following:
$ς z = δ z + y ˜ z ∑ i = 1 j δ i α ˜ z .$

## Author Contributions

Conceptualization, A.A., O.B. and R.A.E.-N.; methodology, A.A., O.B. and R.A.E.-N.; investigation, A.A., O.B. and R.A.E.-N.; resources, A.A., O.B. and R.A.E.-N.; data curation, A.A., O.B. and R.A.E.-N.; writing—original draft preparation, A.A., O.B. and R.A.E.-N.; writing—review and editing, A.A., O.B. and R.A.E.-N.; supervision, A.A., O.B. and R.A.E.-N.; project administration, A.A., O.B. and R.A.E.-N.; funding acquisition, A.A., O.B. and R.A.E.-N. All authors have read and agreed to the published version of the manuscript.

## Funding

This research received no external funding.

Not applicable.

Not applicable.

Not applicable.

## Conflicts of Interest

The authors declare no conflict of interest.

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AlGhamdi, A.; Bazighifan, O.; El-Nabulsi, R.A. Important Criteria for Asymptotic Properties of Nonlinear Differential Equations. Mathematics 2021, 9, 1659. https://doi.org/10.3390/math9141659

AMA Style

AlGhamdi A, Bazighifan O, El-Nabulsi RA. Important Criteria for Asymptotic Properties of Nonlinear Differential Equations. Mathematics. 2021; 9(14):1659. https://doi.org/10.3390/math9141659

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AlGhamdi, Ahmed, Omar Bazighifan, and Rami Ahmad El-Nabulsi. 2021. "Important Criteria for Asymptotic Properties of Nonlinear Differential Equations" Mathematics 9, no. 14: 1659. https://doi.org/10.3390/math9141659

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